Theorem isusp 24200

Description: The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

Ref	Expression
isusp.1	⊢ 𝐵 = (Base‘𝑊)
isusp.2	⊢ 𝑈 = (UnifSt‘𝑊)
isusp.3	⊢ 𝐽 = (TopOpen‘𝑊)

Assertion

Ref	Expression
isusp	⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Proof of Theorem isusp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3480	. 2 ⊢ (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2		0nep0 5328	. . . . 5 ⊢ ∅ ≠ {∅}
3		isusp.1	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑊)
4		fvprc 6868	. . . . . . . . . . . 12 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
5	3, 4	eqtrid 2782	. . . . . . . . . . 11 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅)
6	5	fveq2d 6880	. . . . . . . . . 10 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7		ust0 24158	. . . . . . . . . 10 ⊢ (UnifOn‘∅) = {{∅}}
8	6, 7	eqtrdi 2786	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
9	8	eleq2d 2820	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10		isusp.2	. . . . . . . . . 10 ⊢ 𝑈 = (UnifSt‘𝑊)
11	10	fvexi 6890	. . . . . . . . 9 ⊢ 𝑈 ∈ V
12	11	elsn 4616	. . . . . . . 8 ⊢ (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
13	9, 12	bitrdi 287	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
14		fvprc 6868	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
15	10, 14	eqtrid 2782	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
16	15	eqeq1d 2737	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
17	13, 16	bitrd 279	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
18	17	necon3bbid 2969	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
19	2, 18	mpbiri 258	. . . 4 ⊢ (¬ 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
20	19	con4i 114	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
21	20	adantr 480	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
22		fveq2 6876	. . . . . 6 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
23	22, 10	eqtr4di 2788	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
24		fveq2 6876	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
25	24, 3	eqtr4di 2788	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
26	25	fveq2d 6880	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
27	23, 26	eleq12d 2828	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
28		fveq2 6876	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
29		isusp.3	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
30	28, 29	eqtr4di 2788	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
31	23	fveq2d 6880	. . . . 5 ⊢ (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
32	30, 31	eqeq12d 2751	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)) ↔ 𝐽 = (unifTop‘𝑈)))
33	27, 32	anbi12d 632	. . 3 ⊢ (𝑤 = 𝑊 → (((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤))) ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
34		df-usp 24196	. . 3 ⊢ UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
35	33, 34	elab2g 3659	. 2 ⊢ (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
36	1, 21, 35	pm5.21nii 378	1 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  Vcvv 3459  ∅c0 4308  {csn 4601  ‘cfv 6531  Basecbs 17228  TopOpenctopn 17435  UnifOncust 24138  unifTopcutop 24169  UnifStcuss 24192  UnifSpcusp 24193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-res 5666  df-iota 6484  df-fun 6533  df-fv 6539  df-ust 24139  df-usp 24196

This theorem is referenced by:  ressust  24202  ressusp  24203  tususp  24210  uspreg  24212  ucncn  24223  neipcfilu  24234  ucnextcn  24242  xmsusp  24508